Lecture 7

SoSe24/lec03_sort_alg/algdat_heap.py

@@ -0,0 +1,52 @@
+from SoSe24.algodat.foundation import AlgoDatArray
+
+
+class AlgoDatHeap():
+
+    def __init__(self, values: AlgoDatArray):
+        self.heap = values
+        self.size = len(values)
+
+    def left_child(self, i):
+        return 2*i + 1
+
+    def right_child(self, i):
+        return 2*i + 2
+
+    def parent(self, i):
+        return (i-1)//2
+
+    def max_heapify(self, i):
+        l = self.left_child(i)
+        r = self.right_child(i)
+        largest = i
+        if l <= self.size and self.heap[l-1] > self.heap[i-1]:
+            largest = l
+        if r <= self.size and self.heap[r-1] > self.heap[largest-1]:
+            largest = r
+        if largest != i:
+            self.heap[i-1], self.heap[largest-1] = self.heap[largest-1], self.heap[i-1]
+            self.max_heapify(largest)
+
+    def min_heapify(self, i):
+        l = self.left_child(i)
+        r = self.right_child(i)
+        smallest = i
+        if l <= self.size and self.heap[l-1] < self.heap[i-1]:
+            smallest = l
+        if r <= self.size and self.heap[r-1] < self.heap[smallest-1]:
+            smallest = r
+        if smallest != i:
+            self.heap[i-1], self.heap[smallest-1] = self.heap[smallest-1], self.heap[i-1]
+            self.min_heapify(smallest)
+
+    def build_max_heap(self):
+        for i in range(self.size//2, 0, -1):
+            self.max_heapify(i)
+
+    def build_min_heap(self):
+        for i in range(self.size//2, 0, -1):
+            self.min_heapify(i)
+
+    def as_list(self):
+        return [self.heap[i].value for i in range(self.size)]

SoSe24/lec06_graph/astar.py

@@ -0,0 +1,141 @@
+import math
+from typing import Callable
+import re
+from graph import Graph, AdjacencyListGraph, AdjacencyMatrixGraph, NodeColor, Vertex
+import heapq
+
+def a_star(self, start_name: str, end_name: str, heuristic: Callable[[Vertex], float]):
+    color_map = {}                  # maps vertices to their color
+    distance_map = {}               # maps vertices to their distance from the start vertex
+    predecessor_map = {}            # maps vertices to their predecessor in the traversal tree
+
+    def cost(vertex: Vertex) -> float:
+        """Compute the cost of the path to the given vertex."""
+        if distance_map[vertex] is None:
+            return math.inf
+        return distance_map[vertex] + heuristic(vertex)
+
+    Vertex.__lt__ = lambda self, other: cost(self) < cost(other)
+
+    # Initialize the maps
+    for vertex in self.all_vertices():
+        color_map[vertex] = NodeColor.WHITE
+        distance_map[vertex] = None
+        predecessor_map[vertex] = None
+
+    # Start at the given vertex
+    start_node = self.get_vertex(start_name)
+    color_map[start_node] = NodeColor.GRAY
+    distance_map[start_node] = 0
+
+    # Initialize the queue with the start vertex
+    queue = [] # Use a list instead of a deque, because we need to sort it with
+    heapq.heappush(queue,  start_node)
+
+    # Process the queue
+    while len(queue) > 0:
+        vertex = heapq.heappop(queue)
+        if color_map[vertex] == NodeColor.BLACK:
+            # Skip already processed vertices (possibly with a higher cost)
+            continue
+        if vertex.value == end_name:
+            # Return the distance and predecessor maps
+            return distance_map, predecessor_map
+        for dest, weight in self.get_adjacent_vertices_with_weight(vertex.value):
+            if color_map[dest] == NodeColor.BLACK:
+                continue
+            f = distance_map[vertex] + weight + heuristic(dest)
+            if color_map[dest] == NodeColor.GRAY and f > cost(dest):
+                continue
+            predecessor_map[dest] = vertex
+            distance_map[dest] = distance_map[vertex] + weight
+            heapq.heappush(queue, dest)
+            if color_map[dest] == NodeColor.WHITE:
+                color_map[dest] = NodeColor.GRAY
+        color_map[vertex] = NodeColor.BLACK
+
+    # Return the distance and predecessor maps if no path was found
+    return None, None
+
+
+# Add the a_star method to the Graph classes
+AdjacencyListGraph.a_star = a_star
+AdjacencyMatrixGraph.a_star = a_star
+
+
+if __name__ == "__main__":
+
+    def read_labyrinth_into_graph(graph: Graph, filename: str):
+        """Read a labyrinth from a file into a graph. The file format is a grid of characters:"""
+        start = None
+        end = None
+        with open(filename, "r") as file:
+            nodes = []
+            lines = file.readlines()
+            for y, line in enumerate(lines):
+                for x, char in enumerate(line):
+                    if char in ' AS':
+                        name = pos_to_nodename((x, y))
+                        graph.insert_vertex(name)
+                        nodes.append((x, y))
+                    if char == 'A':
+                        end = (x, y)
+                    if char == 'S':
+                        start = (x, y)
+            for x, y in nodes:
+                name1 = f"x{x}y{y}"
+                for neighbor in [(x - 1, y), (x, y - 1), (x, y + 1), (x + 1, y)]:
+                    if neighbor in nodes:
+                        name_neighbor = pos_to_nodename(neighbor)
+                        graph.connect(name1, name_neighbor, 1)
+        return start, end, lines
+
+
+    def nodename_to_pos(nodename):
+        """Convert a node name to a position (x, y)."""
+        m = re.match(r"(^x(\d*)y(\d*))", nodename)
+        if m:
+            return (int(m.group(2)), int(m.group(3)))
+        return None
+
+
+    def pos_to_nodename(pos):
+        """Convert a position (x, y) to a node name."""
+        x, y = pos
+        return f"x{x}y{y}"
+
+
+    def get_heuristic(end) -> Callable[[Vertex], float]:
+        """Return a heuristic function for the given end position."""
+        def heuristic(v: Vertex) -> float:
+            x, y = nodename_to_pos(v.value)
+            x1, y1 = end
+            # Euclidean distance
+            return math.sqrt(abs(x - x1)**2 + abs(y - y1)**2)
+
+        return heuristic
+
+    def update_lines(lines, distance_map, path):
+        """Update the lines with the path found by the A* algorithm."""
+        def replace_at(x, y, replacement):
+            if lines[y][x] not in " .":
+                return
+            lines[y] = lines[y][:x] + replacement + lines[y][x+1:]
+
+        for node in distance_map.keys():
+            if distance_map[node] is not None:
+                x, y = nodename_to_pos(node.value)
+                replace_at(x, y, ".")
+        for node in path:
+            x, y = nodename_to_pos(node)
+            replace_at(x, y, "*")
+        return lines
+
+    graph = AdjacencyListGraph()
+    #graph = AdjacencyMatrixGraph()
+    start, end, lines = read_labyrinth_into_graph(graph, "../../labyrinth.txt")
+    distance_map, predecessor_map = graph.a_star(pos_to_nodename(start), pos_to_nodename(end), get_heuristic(end))
+    endname = pos_to_nodename(end)
+    lines = update_lines(lines, distance_map, graph.path(endname, predecessor_map))
+    for line in lines:
+        print(line, end="")

SoSe24/lec06_graph/disjoint.py

@@ -0,0 +1,18 @@
+
+
+class DisjointValue():
+
+    def __init__(self, value):
+        self.value = value
+        self.parent = None
+
+    def canonical(self):
+        if self.parent:
+            return self.parent.canonical()
+        return self
+
+    def same_set(self, other):
+        return self.canonical() == other.canonical()
+
+    def union(self, other):
+        self.canonical().parent = other.canonical()

SoSe24/lec06_graph/graph.py

@@ -16,12 +16,15 @@ class Vertex:
     def __init__(self, value):
         self.value = value
 
+    def __repr__(self):
+        return str(self.value)
+
 class Graph:
     """A graph."""
     def insert_vertex(self, name: str):
         raise NotImplementedError("Please implement this method in subclass")
 
-    def connect(self, name1: str, name2: str):
+    def connect(self, name1: str, name2: str, weight: float = 1):
         raise NotImplementedError("Please implement this method in subclass")
 
     def all_vertices(self) -> List[Vertex]:
@@ -33,6 +36,12 @@ class Graph:
     def get_adjacent_vertices(self, name: str) -> List[Vertex]:
         raise NotImplementedError("Please implement this method in subclass")
 
+    def get_adjacent_vertices_with_weight(self, name: str) -> List[tuple[Vertex, float]]:
+        raise NotImplementedError("Please implement this method in subclass")
+
+    def all_edges(self) -> List[tuple[str, str, float]]:
+        raise NotImplementedError("Please implement this method in subclass")
+
     def bfs(self, start_name: str):
         """
         Perform a breadth-first search starting at the given vertex.
@@ -74,6 +83,7 @@ class Graph:
         # Return the distance and predecessor maps
         return distance_map, predecessor_map
 
+
     def path(self, destination, map):
         """
         Compute the path from the start vertex to the given destination vertex.
@@ -100,10 +110,10 @@ class AdjacencyListGraph(Graph):
         if name not in self.adjacency_map:
             self.adjacency_map[name] = []
 
-    def connect(self, name1: str, name2: str):
+    def connect(self, name1: str, name2: str, weight: float = 1):
         adjacency_list = self.adjacency_map[name1]
         dest = self.vertex_map[name2]
-        adjacency_list.append(dest)
+        adjacency_list.append((dest, weight))
 
     def all_vertices(self) -> List[Vertex]:
         return list(self.vertex_map.values())
@@ -112,7 +122,19 @@ class AdjacencyListGraph(Graph):
         return self.vertex_map[name]
 
     def get_adjacent_vertices(self, name: str) -> List[Vertex]:
+        return list(map(lambda x: x[0], self.adjacency_map[name]))
+
+    def get_adjacent_vertices_with_weight(self, name: str) -> List[tuple[Vertex, float]]:
         return self.adjacency_map[name]
+    def all_edges(self) -> List[tuple[str, str, float]]:
+        result = []
+        for name in self.adjacency_map:
+            for (dest, weight) in self.adjacency_map[name]:
+                result.append((name, dest.value, weight))
+        return result
+
+
+
 
 class AdjacencyMatrixGraph(Graph):
     """A graph implemented as an adjacency matrix."""
@@ -126,13 +148,13 @@ class AdjacencyMatrixGraph(Graph):
             self.index_map[name] = len(self.vertex_list)
             self.vertex_list.append(Vertex(name))
             for row in self.adjacency_matrix:   # add a new column to each row
-                row.append(0)
+                row.append(None)
-            self.adjacency_matrix.append([0] * len(self.vertex_list)) # add a new row
+            self.adjacency_matrix.append([None] * len(self.vertex_list)) # add a new row
 
-    def connect(self, name1: str, name2: str):
+    def connect(self, name1: str, name2: str, weight: float = 1):
         index1 = self.index_map[name1]
         index2 = self.index_map[name2]
-        self.adjacency_matrix[index1][index2] = 1
+        self.adjacency_matrix[index1][index2] = weight
 
     def all_vertices(self) -> List[Vertex]:
         return self.vertex_list
@@ -145,11 +167,33 @@ class AdjacencyMatrixGraph(Graph):
         index = self.index_map[name]
         result = []
         for i in range(len(self.vertex_list)):
-            if self.adjacency_matrix[index][i] == 1:
+            if self.adjacency_matrix[index][i] is not None:
                 name = self.vertex_list[i].value
                 result.append(self.get_vertex(name))
         return result
 
+    def get_adjacent_vertices_with_weight(self, name: str) -> List[tuple[Vertex, float]]:
+        index = self.index_map[name]
+        result = []
+        for i in range(len(self.vertex_list)):
+            if self.adjacency_matrix[index][i] is not None:
+                name = self.vertex_list[i].value
+                result.append((self.get_vertex(name), self.adjacency_matrix[index][i]))
+        return result
+
+    def all_edges(self) -> List[tuple[str, str, float]]:
+        result = []
+        for i in range(len(self.vertex_list)):
+            for j in range(len(self.vertex_list)):
+                if self.adjacency_matrix[i][j] is not None:
+                    result.append((self.vertex_list[i].value, self.vertex_list[j].value, self.adjacency_matrix[i][j]))
+        return result
+
+
+
+
+if __name__ == "__main__":
+
     def read_cave_into_graph(graph: Graph, filename: str):
         """Read a cave description from a file and insert it into the given graph."""
         with open(filename, "r") as file:
@@ -169,7 +213,6 @@ def read_cave_into_graph(graph: Graph, filename: str):
                         graph.connect(endnode, startnode)
 
 
-if __name__ == "__main__":
     graph = AdjacencyListGraph()
     #graph = AdjacencyMatrixGraph()
     read_cave_into_graph(graph, "../../hoehle.txt")
@@ -179,3 +222,4 @@ if __name__ == "__main__":
     _, predecessor_map = graph.bfs('Schatzkammer')
     path = graph.path('Höhleneingang', predecessor_map)
     print(path)
+

SoSe24/lec06_graph/mst.py

@@ -0,0 +1,111 @@
+import math
+import re
+
+from SoSe24.lec06_graph.disjoint import DisjointValue
+from graph import Graph, AdjacencyListGraph, AdjacencyMatrixGraph, Vertex
+import heapq
+
+
+def mst_prim(self, start_name: str = None):
+    """ Compute the minimum spanning tree of the graph using Prim's algorithm. """
+
+    distance_map = {}   # maps vertices to their current distance from the spanning tree
+    parent_map = {}     # maps vertices to their predecessor in the spanning tree
+
+    Vertex.__lt__ = lambda self, other: distance_map[self] < distance_map[other]
+
+    queue = []
+
+    if start_name is None:
+        start_name = self.all_vertices()[0].value;
+
+    # Initialize the maps
+    for vertex in self.all_vertices():
+        distance_map[vertex] = 0 if vertex.value == start_name else math.inf
+        parent_map[vertex] = None
+        queue.append(vertex)
+
+    heapq.heapify(queue)    # Convert the list into a heap
+
+    # Process the queue
+    cost = 0 # The cost of the minimum spanning tree
+    while len(queue) > 0:
+        vertex = heapq.heappop(queue)
+        cost += distance_map[vertex] # Add the cost of the edge to the minimum spanning tree
+        for (dest, w) in self.get_adjacent_vertices_with_weight(vertex.value):
+            if dest in queue and distance_map[dest] > w:
+                # Update the distance and parent maps
+                queue.remove(dest)
+                distance_map[dest] = w
+                parent_map[dest] = vertex
+                queue.append(dest)      # Add the vertex back to the queue
+                heapq.heapify(queue)    # Re-heapify the queue
+
+    # Return the distance and predecessor maps
+    return parent_map, cost
+
+
+def mst_kruskal(self, start_name: str = None):
+    """ Compute the minimum spanning tree of the graph using Kruskal's algorithm. """
+
+    cost = 0
+    result = []
+    edges = self.all_edges()
+
+    # Create a disjoint set for each vertex
+    vertex_map = {v.value: DisjointValue(v) for v in self.all_vertices()}
+
+    # Sort the edges by weight
+    edges.sort(key=lambda edge: edge[2])
+
+    # Process the edges
+    for edge in edges:
+        start_name, end_name, weight = edge
+        # Check if the edge creates a cycle
+        if not vertex_map[start_name].same_set(vertex_map[end_name]):
+            result.append(edge)
+            vertex_map[start_name].union(vertex_map[end_name])
+            cost += weight
+
+    return result, cost
+
+
+
+AdjacencyListGraph.mst_prim = mst_prim
+AdjacencyMatrixGraph.mst_prim = mst_prim
+
+AdjacencyListGraph.mst_kruskal = mst_kruskal
+AdjacencyMatrixGraph.mst_kruskal = mst_kruskal
+
+
+def read_elektro_into_graph(graph: Graph, filename: str):
+    pattern = re.compile(r'"([^"]+)";"([^"]+)";(\d+)')
+    with (open(filename, "r") as file):
+        for line in file:
+            m = pattern.match(line)
+            if m:
+                start_name = m.group(1)
+                end_name = m.group(2)
+                cost = int(m.group(3))
+                graph.insert_vertex(start_name)
+                graph.insert_vertex(end_name)
+                graph.connect(start_name, end_name, cost)
+                graph.connect(end_name, start_name, cost)
+
+if __name__ == "__main__":
+
+    graph = AdjacencyMatrixGraph()
+    read_elektro_into_graph(graph, "../../elektro.txt")
+
+    parents, cost = graph.mst_prim()
+    print(f"Kosten nach Prim: {cost}")
+    for node, parent in parents.items():
+        if parent is not None:
+            print(f"{node} - {parent}")
+
+    edges, cost = graph.mst_kruskal()
+    print(f"Kosten nach Kruskal: {cost}")
+    for start_name, end_name, _ in edges:
+        print(f"{start_name} - {end_name}")
+
+

elektro.txt

@@ -0,0 +1,8 @@
+"Höhleneingang";"Ost/West-Passage";5
+"Höhleneingang";"Nord/Süd-Passage";3
+"Nord/Süd-Passage";"Nebelraum";7
+"Steiniger Pfad";"Ost/West-Passage";2
+"Ost/West-Passage";"Schwefelgewölbe";4
+"Schwefelgewölbe";"Steiniger Pfad";1
+"Schatzkammer";"Nebelraum";2
+"Steiniger Pfad";"Schatzkammer";6